Can genetic algorithms be used to learn to play multiple games of the same type?
Genetic algorithms and Neural Networks both are "general" methods, in the sense that they are not "domain-specific", they do not rely specifically on any domain knowledge of the game of Mario. So yes, if they can be used to successfully learn how to play Mario, it is likely that they can also be applied with similar success to other Platformers (or even completely different games). Of course, some games may be more complex than others. Learning Tic Tac Toe will likely be easier than Mario, and learning Mario will likely be easier than StarCraft. But in principle the techniques should be similarly applicable.

If you only want to learn in one environment (e.g., Mario), and then immediately play a different game without separately training again, that's much more complicated. For research in that area you'll want to look for Transfer Learning and/or Multi-Task learning. There has definitely been research there, with the latest developments that I'm aware of having been published yesterday (this is Deep Reinforcement Learning though, no GAs I think).

The most "famous" recent work on training Neural Networks to play games using Genetic Algorithms that I'm aware of is this work by Uber (blog post links to multiple papers). I'm not 100% sure if that really is the state of the art anymore, if it's the best work, etc... I didn't follow all the work on GAs in sufficient detail to tell for sure. It'll be relevant at least though.

I know there's also been quite a lot of work on AI in general for Mario / other platformers (for instance in venues such as the IEEE Conference on Computational Intelligence and Games, and the TCIAIG journal).